| In this thesis we describe intermediate Jacobians of threefolds obtained from singular cubic threefolds. By this we mean two things. First, we describe the intermediate Jacobian of a desingularization of a cubic threefold with isolated singularities. Second, we describe limits of intermediate Jacobians of smooth cubic threefolds, as the family of cubic threefolds acquires isolated singularities. In regards to the first question, generalizing a result of Clemens-Griffiths we show specifically that the intermediate Jacobian of a distinguished desingularization of a cubic threefold with a single singularity of type A3 is the Jacobian of the normalization of an associated complete intersection curve in P3, the so called (2,3)-curve. In regards to degenerations, we describe how the limit intermediate Jacobian, under certain conditions, can be described as a semi-abelian variety as the extension of a torus by the finite quotient of the product of Jacobians of curves, where one of the curves is the normalization of the (2,3)-curve associated to the cubic threefold and a choice of singularity, and the other curves are so-called tails arising from stable reduction of plane curve singularities. |
